Download Sections 2.1/2.2: Sets

Sections 2.1/2.2:
Sets
• A set is a collection of objects.
• An element is an object in the set.
• For a set to be useful, it must be well defined. this means that if a set and an element
are given, it must be possible to determine whether or not the element belongs to the set.
TWO TYPES OF SET NOTATION:
1. Listing method: Listing the elements separated by commas. {Oklahoma, Ohio, Oregon}.
2. Set-Builder Notation: {x | x is a US state which begins with the letter O }.
NOTATION:
• ∈ denotes that an object is in the set.
• ∈
/ denotes that the object is NOT in the set.
• ∅ or {} denote the empty set. The empty set is a set with no elements.
Example 1: Fill in each blank with either ∈ or ∈
/ to make the following statements true.
(a) 4
{2, 4, 6, 8, 10}
(b) 9
{x | x is a multiple of 4}
OTHER DEFINITIONS:
• Cardinality: The number of elements in a set A is called the cardinal number or
cardinality of set A, denoted n(A).
• Equal Sets: Two sets A and B are equal, denoted A = B, if and only if they have the
same elements.
• Equivalent Sets: Two sets A and B are equivalent, denoted A ∼ B, if and only if
they have the same number of elements.
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SECTIONS 2.1/2.2: SETS
• Subset of a set: Set A is said the be a subset of B, denoted A ⊆ B, if and only if
every element of A is also an element of B.
• Proper subset: A is a proper subset of B, denoted A ⊂ B, if A ⊆ B and B has an
element that is not in A.
• Disjoint sets: A and B are disjoint if they have no elements in common.
Example 2: Write true or false for each of the following statements. If false, tell why and
explain how you can correct the statement to make it true.
(a) 6 ∈
/ {3, 7, 8, 9}
(b) {4} ∈ {1, 2, 3, 4}
(c) {1, 2, 3, 4} = {x| x is a counting number less than 5}
(d) {7} ⊂ {1, 7}
(e) {7} ⊆ {1, 7}
(f) {3, 5, 7} ⊂ {3, 5, 7}
• Number of subsets: If a set has n elements, then it has 2n subsets.
• Number of proper subsets: If a set has n elements, then it has 2n −1 proper subsets.
Example 3: Determine the number of subsets for {2, 3, 4}. List all the proper subsets.
Example 4: Determine the number of proper subsets for {a, e, i, o, u}